The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture

TL;DR

This paper advances understanding of the minimum rank of graphs by establishing a weak version of the graph complement conjecture and providing partial results for the delta conjecture using classical and extremal graph theory techniques.

Contribution

It introduces a weak version of the graph complement conjecture applicable to key parameters and offers extremal resolutions for the delta conjecture based on Mader's classical result.

Findings

Established a weak version of the graph complement conjecture for key minimum rank parameters.

Provided extremal resolutions for the delta conjecture using Mader's result.

Improved bounds on the positive semidefinite Colin de Verdière number under certain assumptions.

Abstract

Since the transformative workshop by the American Institute of Mathematics on the minimum rank of a graph, two longstanding open problems have captivated the community interested in the minimum rank of graphs: the graph complement conjecture and the $\delta$-conjecture. In this paper, we use a classical result of Mader (1972) to establish a weak version of the graph complement conjecture for all key minimum rank parameters. In addition, again using the same result of Mader, we present some extremal resolutions of the $\delta$-conjecture. Furthermore, we incorporate the assumption of the $\delta$-conjecture and extensive work on graph degeneracy to improve the bound in the weak version of the graph complement conjecture. We conclude with a list of conjectured bounds on the positive semidefinite variant of the Colin de Verdi\`ere number.